The present invention relates to a method and apparatus for measuring physical properties of a sample of matter, including the mass, volume, density, and bulk modulus. More particularly, the invention relates to a method and apparatus which automatically measures a stress/strain relationship of compressible material.
Several physical properties of matter can be determined from measurements of sample mass, volume, and changes in volume with respect to the specific environment in which the measurements are conducted. For instance, the density of matter may be determined from the mass and volume, as will be briefly described below. Under other testing conditions it is also possible to measure the bulk modulus, or compressibility, of a material by employing some of the same measurement methods.
The density of a substance is expressed as a ratio of mass m to volume V, or m/V. This is a physical property of a material which relates to composition, level of impurities, and mixtures, and can be an indicator of hidden features such as voids. In the case of compressible media, such as closed-pore solids, the bulk density is a function of hydrostatic pressure, since the volume changes but the mass remains constant.
There exist two popular methods for determining the density of a solid: (1) by comparison of the sample density with the densities of substances of known value, usually by hydrostatic weighing in two different fluids of known and substantially different density (Archimedes principle), and (2) by the independent measurement of mass and volume of the sample.
Considering the first method, the weight of an object is measured in two liquids having significantly different densities. The measured weight, also referred to herein as apparent weight, is reduced from the true weight due to buoyancy forces acting on the object. Thus, the apparent weight is the true weight minus the buoyancy force, where the buoyancy force is equal to the weight of the liquid displaced by the object. When in the higher density liquid, the buoyancy force acting on the object is greater, and the apparent weight of the object is less. The sensitivity of the method ultimately relies on the range in available liquid densities. In one method of using hydrostatic weighing techniques, one measures the apparent weight of an object in alcohol and then in water, where the alcohol and the water have densities of 0.791 and 1.0 g/cc, respectively. It is also common practice to weigh the object in air and then in water or alcohol, thus using air as the first medium. In that case it is common practice to assume the weight in air to be the xe2x80x9ctrue weightxe2x80x9d of the object. The specific reasons for using a liquid as one of the mediums in this technique are (i) to obtain a large difference in density between the two fluids, and (ii) to increase the effect of buoyancy forces.
For very accurate measurement of density, there are several experimental problems which are typically ascribed to the hydrostatic weighing method using two liquids. First, the method suffers from the necessity of weighing an object in liquid. Strictly as a practical matter, this requires suspending the object via a tether or thin wire in the liquids. Second, related to the first, is the fact that surface tension forces affect the measurement as the liquid meniscus either pulls the tether down into the liquid or pushes it up into the adjacent gas (typically air), depending on whether the liquid wets the tether easily. Third, the density of the liquid is affected by dissolved gases in the liquid. Since the effect of trapped gas is to change the actual density of the liquid, efforts must be made to eliminate the trapped gas. Fourth, results will vary due to bubbles of trapped gas on irregular sample surfaces of the object. The bubbles that cling due to surface tension displace liquid and affect the measured buoyancy forces.
The second common approach to determine density requires independent determination of both the mass and the volume. One measures the mass of the body using conventional state-of-the-art balances common to most laboratories. Commercial devices exist for performing this step to very high precision and accuracy. The volume is determined independently. If the sample is of a uniform geometry, it may be possible to calculate the specimen volume based on measurable dimensions. In the more general case where samples are of irregular shape, the volume is determined by a method commonly referred to as pycnometry. For reasonably sized samples on the order of 0.5 cubic centimeters and larger, commercial pycnometers are available for determining volume to 0.02%. Pycnometers typically consist of two chambers connected by means of a pathway for a gas to move and a valve which can isolate the two chambers. The exact volume of one of the chambers must be known apriori. The second chamber is of arbitrary, but similar size. The first chamber, of known volume, is pressurized using a gas such as helium to a predetermined pressure. The second chamber is initially empty and is evacuated by means of a vacuum pump. By means of valves, the two chambers are then isolated from the gas source and from the vacuum pump leaving the first chamber at an elevated pressure with helium and the second chamber under vacuum. The valve in the passageway connecting the two chambers is then opened and the pressurized gas is allowed to expand from the first chamber into the second chamber, and the pressure achieves a new equilibrium value by virtue of the increased volume occupied by the gas. It is a straightforward calculation to determine the volume of the unknown chamber using the initial helium pressure in the first chamber, the volume of the first chamber, and the final pressure. The steps are then repeated with the sample of interest being placed into the second chamber. The newly calculated volume of the second chamber represents the remaining volume of the second chamber not occupied by the sample.
Although the field of pycnometry is well established for accurately determining the volumes of solids of reasonable sizes, the state-of-the-art is limited by several factors in attempts to extrapolate to smaller samples. There are many industries where large samples are not always available. Some specific applications would include high-temperature superconducting wires, samples pertaining to the study of irradiation, and porous membranes used for delivering and mixing gases, such as in the fuel cell applications. The volume of such samples is often much smaller than that required by pycnometers. For accurate measurements, the sample should occupy a significant fraction of the chamber volume, e.g. 50-60% of a 1 cubic centimeter chamber.
There are several other limitations to the pycnometry method for determining density. First, if safeguards are not included, temperature variations of the gas due to room temperature fluctuations may affect the pressure, and hence the density, of the pressurized gas. Second, the gas-comparison pycnometer described above does not work if there are any leaks in the system. The ability of the technique to work depends strongly upon the number of gas atoms remaining constant before and after the gas expands into the second chamber. Third, the chamber door, when closed, must close in a repeatable fashion such that the volume of the chamber is exactly the same every time the door is opened and closed. Fourth, the mechanism requires a vacuum pump. Fifth, the true volume of the pressurized chamber must be known to better tolerances than the desired accuracy of sample volume. Finally, the mass must be measured by a separate device.
Further, it is difficult, if not impossible, to determine the bulk modulus of a material by Archimedes method using the conventional procedure briefly described above for at least two reasons. First, the problems identified above with measuring the weight of samples in liquids apply as well in a bulk modulus measurement. Second, to measure bulk modulus, or compressibility, the method must affect a change in volume by a change in hydrostatic stress. As a practical matter, one cannot cause significant changes in stress or volume by simple submersion in a typical laboratory environment. For example, to generate a meager stress of 14 psi on a sample submerged in a column of water, the water column would have to be 34 feet tall.
Thus, there are significant limitations associated with the prior art with respect to density and bulk modulus measurements. The current invention offers significant improvements over the prior art, as will become apparent in the following description of invention.
The foregoing and other needs are met by a method for determining a relationship between hydrostatic stress and volumetric strain in a sample of compressible material, where the sample is immersed in a fluid medium having a density. The method includes the steps of varying the hydrostatic stress on the sample by changing the density of the fluid medium over a range of densities, determining a change in apparent weight of the sample as the density of the fluid medium is changed, and determining the volumetric strain in the sample based on the change in apparent weight of the sample and the change in density of the fluid medium.
An underlying assumption of prior methods for measuring density is that the sample is xe2x80x9cincompressiblexe2x80x9d. By definition, incompressible matter maintains a constant volume with applied stress. While applied stresses always yield some finite strain, most dense engineering materials are practically incompressible.
According to preferred embodiments of the invention, one may determine the bulk modulus, or compressibility, of compressible materials. The bulk modulus describes the volumetric strain when matter is exposed to a hydrostatic stress. The bulk modulus, K, is given by       K    =                  σ                  Δ          ⁢                      xe2x80x83                    ⁢                      V            /                          V              1                                          =              1        B              ,
where "sgr" is the hydrostatic stress, V1 is the initial volume of the sample prior to an applied stress, and xcex94V is the change in volume due to the applied hydrostatic stress. The compressibility, B, is the reciprocal of the bulk modulus.
In preferred embodiments of the invention, the method described above further comprises changing the density of the fluid medium from an nth density, xcfx81n, to an nth+1 density, xcfx81n+1, and determining an nth apparent weight of the sample Wn at the nth density and an nth+1 apparent weight of the sample Wm+1 at the nth+1 density. The volumetric strain in the sample is determined according to:                     Δ        ⁢                  xe2x80x83                ⁢        V                    V        1              =                                        W            n                    -                      W                          n              +              1                                                            V            1                    xc3x97                      (                                          ρ                n                            -                              ρ                                  n                  +                  1                                                      )                              -      1        ,
where V1 is an initial volume of the sample at an initial density, and xcex94V/V1 is the volumetric strain.
In another aspect, the invention provides a method for operating an apparatus to determine bulk modulus of a sample, where the apparatus includes a beam balance enclosed within a pressure chamber, and the beam balance includes a sample pan coupled to a first end and a counter-force application device disposed at an opposing second end. The apparatus also includes a calibration standard of known density, a temperature sensor, and a pressure sensor. The method includes the steps of sealing the sample within the pressure chamber, purging the pressure chamber of air, and pressurizing the chamber to an nth pressure using a fluid medium. With the sample on the sample pan and the pressure chamber pressurized to the nth pressure, an nth force is determined, where the nth force is applied by the counter-force application device as required to balance the beam balance. Using the temperature sensor and the pressure sensor, the nth temperature and the nth pressure within the pressure chamber are measured. Based upon a predetermined relationship between density, pressure, and temperature of the fluid medium, an nth density of the fluid medium is calculated.
The method further includes pressurizing the chamber to an nth+1 pressure, which is different from the nth pressure. With the sample on the sample pan and the pressure chamber pressurized to the nth+1 pressure, an nth+1 force required to balance the beam balance is determined, which is applied by the counter-force application device. Using the temperature sensor and the pressure sensor, the nth+1 temperature and the nth+1 pressure within the pressure chamber are measured. Based upon the predetermined relationship between density, pressure, and temperature of the fluid medium, an nth+1 density of the fluid medium is calculated. The bulk modulus of the sample is then calculated based at least in part on the nth and nth+1 densities of the fluid medium and the nth and nth+1 forces applied by the counter-force application device.
In yet another aspect, the invention provides an apparatus for determining bulk modulus of a sample that is immersed in a fluid medium having variable density while exposed to an acceleration in a first direction and a net buoyancy force in a second direction opposite the first direction, where the net buoyancy force is a sum of buoyancy forces in the first and second directions exerted on the sample by the fluid medium. The apparatus comprises a chamber for containing the fluid medium and the sample immersed in the fluid medium, and means for selectively varying the density of the fluid medium in the chamber over a range of densities. The apparatus also includes means for producing at least one electrical signal related to the density of the fluid medium in the chamber as the density of the fluid medium is varied.
A beam balance, having opposing first and second ends, is disposed within the chamber. The beam balance includes a sample pan disposed adjacent the first end of the beam balance. The sample pan has a sample pan volume and a sample pan mass, and creates a sample pan moment adjacent the first end of the beam balance. A first counter-weight is disposed adjacent the second end of the beam balance. The first counter-weight has a first counter-weight volume which is substantially equivalent to the sample pan volume, and a first counter-weight mass which is substantially equivalent to the sample pan mass. Thus, the first counter-weight creates a first counter-weight moment adjacent the second end of the beam balance that is substantially equivalent to the sample pan moment.
A coil assembly, which is disposed adjacent the second end of the beam balance, has a coil assembly volume and a coil assembly mass, and creates a coil assembly moment adjacent the second end of the beam balance. The coil assembly is electrically coupled to a controller. A magnet assembly is disposed adjacent to and magnetically interacts with the coil assembly.
A second counter-weight, which is disposed adjacent the first end of the beam balance, has a second counter-weight volume which is substantially equivalent to the coil assembly volume, and a second counter-weight mass which is substantially equivalent to the coil assembly mass. The second counter-weight creates a second counter-weight moment adjacent the first end of the beam balance that is substantially equivalent to the coil assembly moment.
The controller provides a coil current to the coil assembly, thereby generating a magnetic field that interacts with the magnet assembly. The interaction between the magnetic field of the coil assembly and the magnet assembly causes a force to be applied to the second end of the beam to keep the beam balanced as the density of the fluid medium in the chamber is varied over the range of densities. The force applied to the second end of the beam is substantially equivalent to the difference between the net buoyancy force and the product of the sample mass times the acceleration while the sample is immersed in the fluid medium as the density of the gaseous medium is varied over the range of densities.
A computing device receives the at least one electrical signal related to the density of the fluid medium and at least one electrical signal related to the coil current, and based thereon calculates the bulk modulus of the sample.